A Brief History of Quaternions

Complex numbers were a hot subject for research in the early eighteen hundreds.
An obvious question was that if a rule for multiplying two numbers together was
known, what about multiplying three numbers? For over a decade, this simple
question had bothered Hamilton, the big mathematician of his day. The pressure
to find a solution was not merely from within. Hamilton wrote to his son:

Every morning in the early part of the above-cited month [Oct. 1843] on my
coming down to breakfast, your brother William Edwin and yourself used to ask
me, 'Well, Papa, can you multiply triplets?' Whereto I was always obliged to
reply, with a sad shake of the head, 'No, I can only add and subtract them.'

We can guess how Hollywood would handle the Brougham Bridge scene in Dublin.
Strolling along the Royal Canal with Mrs. H-, he realizes the solution to the
problem, jots it down in a notebook. So excited, he took out a knife and
carved the answer in the stone of the bridge.

Hamilton had found a long sought-after solution, but it was weird, very weird,
it was 4D. One of the first things Hamilton did was get rid of the fourth
dimension, setting it equal to zero, and calling the result a "proper
quaternion." He spent the rest of his life trying to find a use for
quaternions. By the end of the nineteenth century, quaternions were viewed as
an oversold novelty.

In the early years of this century, Prof. Gibbs of Yale found a use for proper
quaternions by reducing the extra fluid surrounding Hamilton's work and adding
key ingredients from Rodrigues concerning the application to the rotation of
spheres. He ended up with the vector dot product and cross product we know
today. This was a useful and potent brew. Our investment in vectors is
enormous, eclipsing their place of birth (Harvard had >1000 references under
"vector", about 20 under "quaternions", most of those written before the turn
of the century).

In the early years of this century, Albert Einstein found a use for four
dimensions. In order to make the speed of light constant for all inertial
observers, space and time had to be united. Here was a topic tailor-made for a
4D tool, but Albert was not a math buff, and built a machine that worked from
locally available parts. We can say now that Einstein discovered Minkowski
spacetime and the Lorentz transformation, the tools required to solve problems
in special relativity.

Today, quaternions are of interest to historians of mathematics. Vector
analysis performs the daily mathematical routine that could also be done with
quaternions. I personally think that there may be 4D roads in physics that can
be efficiently traveled only by quaternions, and that is the path which is laid
out in these web pages.

In a longer history, Gauss would get the credit for seeing quaternions first in
one of his notebooks. Rodrigues developed 3D rotations all on his own also in
the 1840's. The Pauli spin matrices and Penrose's spinors are reinventions of
the wheel that miss out on division. Although I believe that is a major
omission and cause of subtle flaws at the foundations of modern physics, spin
matrices and spinors have many more adherents today than quaternions.